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Electromagnetic Induction

- Chapter 22

Expectations

- After this chapter, students will
- Calculate the EMF resulting from the motion of

conductors in a magnetic field - Understand the concept of magnetic flux, and

calculate the value of a magnetic flux - Understand and apply Faradays Law of

electromagnetic induction - Understand and apply Lenzs Law

Expectations

- After this chapter, students will
- Apply Faradays and Lenzs Laws to some

particular devices - Electric generators
- Electrical transformers
- Calculate the mutual inductance of two conducting

coils - Calculate the self-inductance of a conducting

coil

Motional EMF

- A wire passes through a uniform magnetic field.

The length of the wire, the magnetic field, and

the velocity of the wire are all perpendicular to

one another

Motional EMF

- A positive charge in the wire experiences a

magnetic force, directed upward

Motional EMF

- A negative charge in the wire experiences the

same magnetic force, but directed downward - These forces tend to separate the charges.

Motional EMF

- The separation of the charges produces an

electric field, E. It exerts an attractive force

on the charges

E

Motional EMF

- In the steady state (at equilibrium), the

magnitudes of the magnetic force separating

the charges and the Coulomb force attracting

them are equal.

E

Motional EMF

- Rewrite the electric field as a potential

gradient - Substitute this result back into our earlier

equation

E

Motional EMF

- Substitute this result back into our earlier

equation

E

Motional EMF

- This is called motional EMF. It results from the

constant velocity of the wire through the

magnetic field, B.

E

Motional EMF

- Now, our moving wire slides over two other wires,

forming a circuit. A current will flow, and

power is dissipated in the resistive load

Motional EMF

- Where does this power come from?
- Consider the magnetic
- force acting on the
- current in the sliding
- wire

Motional EMF

- Right-hand rule 1 shows that this force opposes

the motion of the wire. To move the wire at

constant velocity requires an equal and opposite

force. - That force does work
- The power

Motional EMF

- The forces magnitude was calculated as
- Substituting
- which is the same as the
- power dissipated electrically.

Motional EMF

- Suppose that, instead of being perpendicular to

the plane of the sliding-wire circuit, the

magnetic field had made an angle f with the

perpendicular to that plane. - The perpendicular
- component of B B cos f

Motional EMF

- The motional EMF
- Rewrite the velocity
- Substitute

Motional EMF

- L Dx is simply the change in the loop area.

Motional EMF

- Define a quantity F
- Then
- F is called magnetic
- flux.
- SI units Tm2 Wb (Weber)

Magnetic Flux

- Wilhelm Eduard Weber
- 1804 1891
- German physicist and mathematician

Faradays Law

- In our previous result, we said that the induced

EMF was equal to the time rate of change of

magnetic flux through a conducting loop. This,

rewritten slightly, is called Faradays Law - Why the minus sign?

Faradays Law

- Michael Faraday
- 1791 1867
- English physicist
- and mathematician

Faradays Law

- To make Faradays Law complete, consider adding N

conducting loops (a coil) - What can change the magnetic flux?
- B could change, in magnitude or direction
- A could change
- f could change (the coil could rotate)

Lenzs Law

- Here is where we get the minus sign in Faradays

Law - Lenzs Law says that the direction of the induced

current is always such as to oppose the change in

magnetic flux that produced it. - The minus sign in Faradays Law reminds us of

that.

Lenzs Law

- Heinrich Friedrich Emil Lenz
- 1804 1865
- Russian physicist

Lenzs Law

- Lenzs Law says that the direction of the induced

current is always such as to oppose the change in

magnetic flux that produced it. - What does that mean?
- How can an induced current oppose a change in

magnetic flux?

Lenzs Law

- How can an induced current oppose a change in

magnetic flux? - A changing magnetic flux induces a current.
- The induced current produces a magnetic field.
- The direction of the induced current determines

the direction of the magnetic field it produces. - The current will flow in the direction (remember

right-hand rule 2) that produces a magnetic

field that works against the original change in

magnetic flux.

Faradays Law the Generator

- A coil rotates with a constant angular speed in a

magnetic field. - but f changes
- with time

Faradays Law the Generator

- So the flux also changes with time
- Get the time rate of change (a calculus problem)
- Substitute into Faradays Law

Faradays Law the Generator

- The maximum voltage occurs when
- What makes the voltage larger?
- more turns in the coil
- a larger coil area
- a stronger magnetic field
- a larger angular speed

Back EMF in Electric Motors

- An electric motor also contains a coil rotating

in a magnetic field. - In accordance with Lenzs Law, it generates a

voltage, called the back EMF, that acts to oppose

its motion.

Back EMF in Electric Motors

- Apply Kirchhoffs loop rule

Mutual Inductance

- A current in a coil produces a magnetic field.
- If the current changes, the magnetic field

changes. - Suppose another coil is nearby. Part of the

magnetic field produced by the first coil

occupies the inside of the second coil.

Mutual Inductance

- Faradays Law says that the changing magnetic

flux in the second coil produces a voltage in

that coil. - The net flux in the
- secondary

Mutual Inductance

- Convert to an equation, using a constant of

proportionality

Mutual Inductance

- The constant of proportionality is called the

mutual inductance

Mutual Inductance

- Substitute this into Faradays Law
- SI units of mutual inductance Vs / A henry (H)

Mutual Inductance

- Joseph Henry
- 1797 1878
- American physicist

Self-Inductance

- Changing current in a primary coil induces a

voltage in a secondary coil. - Changing current in a coil also induces a voltage

in that same coil. - This is called self-inductance.

Self-Inductance

- The self-induced voltage is calculated from

Faradays Law, just as we did the mutual

inductance. - The result
- The self-inductance, L, of a coil is also

measured in henries. It is usually just called

the inductance.

Mutual Inductance Transformers

- A transformer is two coils wound around a common

iron core.

Mutual Inductance Transformers

- The self-induced voltage in the primary is
- Through mutual induction, and EMF appears in the

secondary - Their ratio

Mutual Inductance Transformers

- This transformer equation is normally written
- The principle of energy conservation requires

that the power in both coils be equal (neglecting

heating losses in the core).

Inductors and Stored Energy

- When current flows in an inductor, work has been

done to create the magnetic field in the coil.

As long as the current flows, energy is stored in

that field, according to

Inductors and Stored Energy

- In general, a volume in which a magnetic field

exists has an energy density (energy per unit

volume) stored in the field